Election lawyers typically allow such groups to keep their donors secret as long as 51 percent of the group's activities aren't related to politics and the group does not coordinate with candidates, CNBC noted.

Lewandowski, a senior adviser to the group, met with Trump at the White House ahead of the president's speech at the Conservative Political Action Conference (CPAC) last week.

"Our primary purpose has to be issue-related, but that doesn't mean we can't throw in questions that are news-related or political [in our polling]," Erin Montgomery, the group's communications director, told CNBC.

"We share our polling with lots of outside groups and elected officials, and we put all of them up online," said Montgomery. "I don't know if this information is ever shared with the White House. Anyone can see it."

This week, Trump also announced that Brad Parscale, one of the group's founding members, would lead his 2020 reelection campaign. Parscale previously led Trump's digital operation during the 2016 campaign.

Montgomery and the group's executive director, Brian Walsh, told CNBC the group operates with "high walls and clear lines" between itself and the White House and Republican National Committee (RNC).

"In the old world [of political polling], things were kept secret. In the new world, you make everything public," Walsh said.

Critics say the group is performing work typically done by national political parties or campaigns, and is not operating with clear enough distinction from those organizations while enjoying the benefits of a secretive donor list.

"[America First Policies] is doing the type of polling that would typically be done by a presidential campaign or a major party committee like the RNC or the DNC," said Brendan Fischer of the nonprofit Campaign Legal Center.

"So even though they claim to be committed to a set of issues, the available evidence here indicates that they're operating as a polling shop for the president."